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An extension of the Fuglede–Putnam theorem to \((p,k)\)-quasihyponormal operators. (English) Zbl 1101.47024

A bounded operator \(T\) acting on some complex Hilbert space \(H\) is said to be \((p,k)\)-quasihyponormal for some \(0<p\leq 1\) and \(k\in \mathbb{N}\) if \(A^{*k}[(A^*A)^p-(AA^*)^p]A^k\geq 0\). In the present paper, the author shows that if \(A\) and \(B^*\) are \((p,k)\)-quasihyponormal operators with \(B\) invertible and if \(AX=XB\) for some Hilbert–Schmidt operator \(X\), then \(A^*X=XB^*\). The paper also contains several applications of this result related to the generalized derivations \(\delta_{A,B}\) induced by these type of operators on the space of Hilbert–Schmidt operators.

MSC:

47B47 Commutators, derivations, elementary operators, etc.
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47B20 Subnormal operators, hyponormal operators, etc.
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